Proving the Special Adjoint Functor Theorem from the general Adjoint Functor Theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:09:29Z http://mathoverflow.net/feeds/question/25793 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25793/proving-the-special-adjoint-functor-theorem-from-the-general-adjoint-functor-theo Proving the Special Adjoint Functor Theorem from the general Adjoint Functor Theorem Miha Habič 2010-05-24T19:19:39Z 2010-05-27T08:00:51Z <p>Often, when dealing with adjoint functor theorems, people go about proving each one separately, from first principles if you will (this is the course taken in MacLane). However, the names suggest there is a deeper connection between the two. </p> <p>These are the relevant theorems:</p> <ul> <li><p><strong>AFT:</strong> For $D$ a complete locally small category, a functor $G:D\to C$ has a left adjoint iff it preserves small limits and satisfies the solution set condition (for each object $c$ in $C$ we have a weakly universal set from $c$ to $G$)</p></li> <li><p><strong>SAFT (general form):</strong> Let $D$ be a complete locally small category with a cogenerating set and let there exist pullbacks of classes of subobjects of objects in $D$. Let $C$ be locally small. A functor $G:D \to C$ has a left adjoint iff it preserves small limits and pullbacks of classes of monics.</p></li> <li><p><strong>SAFT (classical form):</strong> Let $D$ be a complete, well-powered, locally small category with a cogenerating set. Let $C$ be locally small. A functor $G:D \to C$ has a left adjoint iff it preserves small limits.</p></li> </ul> <p>I wonder if it is possible to deduce SAFT from AFT; can we, given the hypotheses of SAFT, in some way apply AFT?</p> <p>Of course this is trivially true. One simply goes through the standard proof of SAFT, at some point forgets what he is doing, produces some universal arrows and applies AFT. This isn't really interesting.</p> <p>What is interesting is that we can produce a nontrivial proof of SAFT in the case that the categories involved are well-powered (so we can nontrivially prove the classical form of SAFT). You can, broadly speaking, get a solution set at an object $c$ as the set of arrows of the form $c \to G r$, where $r$ ranges over the subobjects of a certain product of the elements of the cogenerating set. But the proof doesn't translate to a general setting.</p> <p>My question is this: can AFT be applied in a nontrivial way to prove the general form of SAFT? Based on the hypotheses of SAFT, can we construct nontrivial solution sets?</p> <p>I'm thinking we can't. To construct a solution set, we need to already have some set, from which we can then take elements, and local smallness doesn't seem to be enough here. Also, MacLane doesn't mention this variation on the proof, but does mention the classical case. </p>